Abstract
A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ. If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.
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References
Abate, J., Whitt, W.: Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J. Comput. 11, 394–405 (1999)
Altman, E., Yechiali, U.: Analysis of customer’s impatience in queues with server vacations. Queueing Syst. 52, 261–279 (2006)
Bowman, K.O., Shenton, L.R.: Continued Fractions in Statistical Applications. Dekker, New York (1989)
Chakravarthy, S.R.: A disaster queue with Markovian arrivals and impatient customers. Appl. Math. Comput. 214, 48–59 (2009)
Chen, A., Renshaw, E.: The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Probab. 34(1), 192–207 (1997)
Gelenbe, E.: Product form networks with negative and positive customers. J. Appl. Probab. 28, 655–663 (1991)
Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 6th edn. Academic Press, New York (2000)
Jain, G., Sigman, K.: A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Probab. 33(4), 1191–1200 (1996)
Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. Studies in Computational Mathematics, vol. 3. Elsevier, Amsterdam (1992)
Parthasarathy, P.R., Lenin, R.B.: Birth and Death Process (BDP) Models with Applications—Queueing, Communication Systems, Chemical Models, Biological Models: The State-of-the-Art with a Time-Dependent Perspective. American Series in Mathematical and Management Sciences, vol. 51. American Sciences Press, Columbus (2004)
Parthasarathy, P.R., Selvaraju, N.: Transient analysis of a queue where potential customers are discouraged by queue length. Math. Probl. Eng. 7, 433–454 (2001)
Parthasarathy, P.R., Sudhesh, R.: Exact transient solution of state-dependent birth-death processes. J. Appl. Math. Stoch. Anal. 2006, 1–16 (2006). Article ID 97073
Parthasarathy, P.R., Sudhesh, R.: Exact transient solution of a discrete time queue with state-dependent rates. Am. J. Math. Manag. Sci. 26, 253–276 (2006)
Parthasarathy, P.R., Sudhesh, R.: Time-dependent analysis of a single-server retrial queue with state-dependent rates. Oper. Res. Lett. 35, 601–611 (2007)
Yechiali, U.: Queues with system disasters and impatient customers when system is down. Queueing Syst. 56, 195–202 (2007)
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Sudhesh, R. Transient analysis of a queue with system disasters and customer impatience. Queueing Syst 66, 95–105 (2010). https://doi.org/10.1007/s11134-010-9186-x
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DOI: https://doi.org/10.1007/s11134-010-9186-x
Keywords
- Single server queue
- Server breakdown
- Impatient customers
- Transient probabilities
- Continued fractions
- Confluent hypergeometric functions